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This problem involves a normal distribution with \(X \sim N(1, t^2)\), and requires two separate calculations involving probabilities.

### Part 1:
We are asked to determine values of \(t\) such that:
\[
P(X \leq 5t) \geq \frac{1}{2}
\]
Given that \(X\) follows a normal distribution \( N(1, t^2)\), this means the mean of the distribution is \(1\), and the variance is \(t^2\).

We convert this inequality into the standard normal form by using the standardization formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \(Z \sim N(0, 1)\). In this case, \( \mu = 1 \) and \( \sigma = t \), so:
\[
P \left( \frac{5t - 1}{t} \right) = P \left( Z \leq \frac{5t - 1}{t} \right)
\]
Simplifying:
\[
P \left( Z \leq 5 - \frac{1}{t} \right) \geq \frac{1}{2}
\]
From standard normal tables, we know \(P(Z \leq 0) = \frac{1}{2}\), so we require:
\[
5 - \frac{1}{t} \geq 0 \implies t \geq \frac{1}{5}
\]
Thus, the condition holds for all \(t \geq \frac{1}{5}\).

### Part 2:
For the second part, the problem asks us to find \(k\) for the probability:
\[
P(t^2 - t + 1 \leq X \leq t^2 + t + 1)
\]
We again convert this into a standard normal form. Using the transformation \(Z = \frac{X - 1}{t}\), the probability becomes:
\[
P \left( \frac{(t^2 - t + 1) - 1}{t} \leq Z \leq \frac{(t^2 + t + 1) - 1}{t} \right)
\]
which simplifies to:
\[
P \left( \frac{t^2 - t}{t} \leq Z \leq \frac{t^2 + t}{t} \right) = P(t - 1 \leq Z \leq t + 1)
\]
Now, using the provided standard normal table, we calculate this probability for the maximum \(t\), say \(t = 1.2\), to find \(k\).

From the table, we have:
\[
P(0 \leq Z \leq 1.2) = 0.385
\]
Thus, \(k = 0.385\), and the final answer is:
\[
1000 \times k = 1000 \times 0.385 = 385
\]

### Final Answer:
The value of \(1000 \times k = 385\).

---

Do you want more details on any step or have further questions?

### Related Questions:
1. How do we standardize a normal distribution?
2. What is the standard normal table used for in probability?
3. Why is \(P(Z \leq 0) = \frac{1}{2}\)?
4. How do you interpret the result of a probability inequality like this one?
5. What are some applications of normal distributions in real-world problems?

**Tip**: Standardization of variables is crucial when working with different distributions since it allows us to use a universal probability table (like the standard normal table).

양수 t에 대하여 확률변수 X가 정규분포 N(1, t^2)을 따른다. P(X ≤ 5t) ≥ 1/2를 만족시키는 모든 양수 t에 대하여 P(t^2 - t + 1 ≤ X ≤ t^2 + t + 1)의 최대값을 구하시오.

This page explains how to solve a normal distribution problem where X follows a N(1, t^2) distribution. Learn how to standardize the variable and calculate the probabilities using the cumulative distribution function (CDF). Suitable for undergraduate students.

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